Abstract
We expand the Askey-Wilson (AW) density in a series of products of continuous q-Hermite polynomials times the density that makes these polynomials orthogonal. As a by-product we obtain the value of the AW integral as well as the values of integrals of q-Hermite polynomial times the AW density (q-Hermite moments of AW density). Our approach uses nice, old formulae of Carlitz and is general enough to venture a generalization. We prove that it is possible and pave the way how to do it. As a result, we obtain a system of recurrences that if solved successfully gives a sequence of generalized AW densities with more and more parameters. MSC: 33D45, 05A30, 05E05.
Highlights
1 Introduction and preliminaries 1.1 Introduction We consider a sequence of nonnegative, integrable functions gn : [, ] → R+ defined by the formula n gn x|a(n), q = fh(x|q) φh(x|aj, q), j=
Where a(n) = (a, . . . , an), functions fh and φh defined by ( . ) and ( . ) denote respectively the density of measure that makes the so-called continuous q-Hermite polynomials orthogonal and the generating function of these polynomials calculated at points aj, j =, . . . , n
Similar observations were made in [ ] when commenting on formula ( . . )
Summary
) denote respectively the density of measure that makes the so-called continuous q-Hermite polynomials orthogonal and the generating function of these polynomials calculated at points aj, j = , . Tj(n)(a(n) (q)j hj (x|q), where {hn} denote q-Hermite polynomials, {Tj(n)} are the sequences of certain symmetric functions and {An} are the values of the integrals gn x|a(n), q dx, Hn+ (x|q) = xHn(x|q) – [n]qHn– (x|q), which shows that Hn(x|q) = Hn(x), where {Hn} denote the so-called probabilistic Hermite polynomials, i.√e., polynomials orthogonal with respect to the measure with density equal to exp(–x / )/ π .
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